Fundamental Theorem of Algebra - A Rough Sketch of its Derivation

The fundamental theorem of algebra connects the number of roots a polynomial has in the complex plane to the degree of the highest exponent.  Like if the highest exponent you have in the polynomial is four, then you can expect there to be four roots of the polynomial in the complex plane.  Here's a rough argument for why this happens to be true.  I definitely do not promise that it is error-free.

First of all, since derivatives are preserved under addition and multiplication, polynomials are always complex differentiable.  They have a derivative everywhere.  And because they have a derivative everywhere, the Cauchy-Riemann equations are applicable.  

These equations cancel out nicely when you do a contour integral on a complex surface.  So this makes the value at the center of a circle integral equal the average of the values of the circumference.

Now, unless the function is unbounded, this mean value contour property is exactly what leads to Liouville's theorem in complex analysis which says that every bounded function with a derivative everywhere must be constant.  

We can use this theorem to show by contradiction that every non-constant polynomial contains at least one root.  

Assume it's non-constant, and arrive at a contradiction.

Suppose we have a non-constant polynomial with no roots.  These polynomials are unbounded like all polynomials.  The reciprocal of this polynomial will be bounded though.  When the original polynomial goes to infinity, the reciprocal will go to zero.  So it will be bounded instead.  So this reciprocal is bounded, non-constant, and have a derivative everywhere.  But by Liouville's theorem, since it's bounded and has a derivative everywhere, it should be constant.  Now we have a contradiction from assuming the polynomial was non-constant with zero roots.  Therefore it must be non-constant and possess at least one root.

So we cannot claim a non-constant polynomial contains no roots.  It must contain at least one.

So for whatever non-constant polynomial we have, we can divide it by x minus its root, (x-r), since we know it has at least one of them.  We can repeat this process until we have no roots left.  We will end up with some product of (x-r)'s, and the number of them will equal the degree of the polynomial.  That's the fundamental theorem of algebra.

The most interesting thing to me about this is the restrictive properties of fully differentiable functions on the complex plane. They seem to be much more restrictive than differentiable functions on the reals.